The main goal of this article is to explore the concepts graded ϕ-2-absorbing and primary submodules as a new generalization 2-absorbing submodules. Let ϕ:GS(M)→GS(M)⋃{∅} be function, where GS(M) denotes collection R-submodules M. A proper K∈GS(M) said R-submodule M if whenever x,y are homogeneous elements R s element with xys∈K−ϕ(K), then xs∈K or ys∈K xy∈(K:RM), we call K xs ys in radical xy∈(...

let r be a commutative ring with non-zero identity and m be a unital r-module. then the concept of quasi-secondary submodules of m is introduced and some results concerning this class of submodules is obtained

Secondary homotopy groups supplement the structure of classical homotopy groups. They yield a 2-functor on the groupoid-enriched category of pointed spaces compatible with fiber sequences, suspensions and loop spaces. They also yield algebraic models of (n− 1)-connected (n+1)-types for n ≥ 0. Introduction The computation of homotopy groups of spheres in low degrees in [Tod62] uses heavily secon...

For a triple (G,A, κ) (where G is a group, A is a G-module and κ : G → A is a 3-cocycle) and a G-module B we introduce a new cohomology theory 2Hn(G,A, κ;B) which we call the secondary cohomology. We give a construction that associates to a pointed topological space (X, x0) an invariant 2κ 4 ∈ 2H (π1(X), π2(X), κ; π3(X)). This construction can be seen a “3type” generalization of the classical k...